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\section{Detailed Proof for Discrete Maximum Principle}\par
(以下证明需要对connectedness这个定义进行加强,并且这个加强是必须的)

For convenience, we call the set of all points related to $L_hU_P$ as a $P-$stencil, i.e.,
a $P-$stencil is $\{P\}\cup Q$ supposing $L_hU_P=c_PU_P-\sum_Qc_QU_Q$. Denote $M_\Omega:=\max_{p\in X_\Omega}U_p$ and $M_{\partial\Omega}:=\max_{p \in X_{\partial\Omega}}U_p$.

First, suppose a $p-$stencil is composed of pure Interior Point and we call it \textbf{Pure}. Then if $U_{p}=M_\Omega$, all points in the pure ${p}-$stencil will equal $M_\Omega$.
\begin{proof}[Proof for Lemma 1.]
   This is because
$$
M_\Omega=U_{P}\leq \frac{1}{c_{P}}\sum_{Q}c_{Q}U_{Q}\leq (\frac{1}{c_{P}}\sum_{Q}c_{Q})M_\Omega \leq M_\Omega,
$$
which yields $U_{Q}=M_\Omega$. 
\end{proof}

Next, suppose a $p-$stencil is composed of both Boundray Points and Interior Points. 
And we call it \textbf{Mixed}. 
Then if $U_{p}=M_\Omega$ for a mixed $p-$stencil, we will directly have $M_\Omega\leq M_{\partial\Omega}$. 
\begin{proof}[Proof for Lemma 2.]
To prove it, denote $\widehat{Q}$ as Boundray Point Set in $Q$ and thus 
$Q\backslash\widehat{Q}$ is Interior Point Set.
Since $c_pU_p \leq \sum_{r\in Q\backslash \widehat{Q}}c_rU_r+\sum_{\widehat{Q}}c_{\widehat{Q}}U_{\widehat{Q}}$, $U_p=M_\Omega$, and $U_{r}\leq M_\Omega$, we have
\begin{align*}
&c_pM_\Omega \leq (\sum_{Q\backslash \widehat{Q}}c_r)M_\Omega+\sum_{\widehat{Q}}c_{\widehat{Q}}U_{\widehat{Q}},\Rightarrow \\
&(c_p-\sum_{Q\backslash\widehat{Q}}c_r)M_\Omega \leq \sum_{\widehat{Q}}c_{\widehat{Q}}U_{\widehat{Q}}\leq (\sum_{\widehat{Q}}c_{\widehat{Q}}) M_{\partial\Omega}.\\
&\text{Also, $\frac{c_p-\sum_{Q/\widehat{Q}}c_r}{\sum_{\widehat{Q}}c_{\widehat{Q}}}\geq1$ and $M_\Omega$ is non-negative, }\\
&\text{hence }1\leq\frac{c_p-\sum_{Q/\widehat{Q}}c_r}{\sum_{\widehat{Q}}c_{\widehat{Q}}}\leq \frac{M_{\partial\Omega}}{M_\Omega}, \;(M_\Omega \neq 0) 
\end{align*}   
 and thus  $M_\Omega\leq M_{\partial\Omega}$. 
\end{proof}

Therefore, it suffices to show there indeed exists a mixed $p-$stencil with $U_p=M_\Omega$. By definition of connectedness in Book, however, such a stencil may not exist.

Then we adjust the connectedness to be the version in Class:

$X_\Omega$ is \textbf{connected}, i.e., $\forall\;P_0,P_{m}\in X_\Omega$, there exists a sequence of $P_1,\cdots,P_{m-1}$ such that $\forall r \in 1,\cdots,m,$ ${P_r}$ is in the $P_{r-1}-$stencil. 

Finally we prove the existence of mixed $p-$stencil with $U_p=M_\Omega$ in such sense of connectedness. 
\begin{proof}[Proof.]
We use the proof of contradiction. Suppose a mixed $p-$stencil with $U_p=M_{\Omega}$ never exists. 
By (DMP-4), there exists a $t \in X_{\Omega}$ that its stencil involves at least one boundray point. Hence $t-$stencil is mixed. Then pick $P\in X_{\Omega}$ such that $U_P=M_{\Omega}$. Since $X_\Omega$ is connected, for $P$ and $t$, we connect them with stencils of $P_i$, $i=1,\cdots,m$. For $P$, if $P-$stencil is mixed, by $U_P=M_{\Omega}$ we construct a contradiction. 
Hence it has to be pure and $P_1 \in P-$stencil yields $U_{P_1}=M_{\Omega}$ due to \textbf{Lemma 1}. For $P_1$, if $P_1-$stencil is mixed, by $U_{P_1}=M_{\Omega}$ we construct a contradition. Hence it has to be pure and $U_{P_2}=M_{\Omega}$. Therefore, just repeating it for $m$ times and all $P_i-$ stencil is pure with $U_{P_i}=M_{\Omega}$ for $i=1,\cdots,m$. 
Then $U_t=M_{\Omega}$ and thus the $t-$stencil is an contradition.  
\end{proof}
The existence of a mixed $p-$stencil with $U_{p}=M_{\Omega}$ and \textbf{Lemma 2} yield $M_\Omega\leq M_{\partial\Omega}$.   

\newpage

\section*{Remark 1}
Notation 4分别定义了$X_{\partial\Omega}$和$X_\Omega$作为boundray point set和interior point set,但这里的两个集合并不是常识中的边界与内点。实际上,boundray point set是只由Dirichlet条件控制的边界点。interior point set不仅是区域内部的点，还要再加上非Dirichlet条件(例如Neumann type,General type)控制的边界点。

这种奇怪的叫法或许可以这么理解。$X_{\partial\Omega}$必须是precondition中已经赋值的区域。作为常量,它们与之后的求解内部点无关,这种才算"真边缘人"。$X_\Omega$中的一部分是区域内部的点,另一部分虽然是边界上的点，但由于它们的值也要依赖于内部的点，也要参与内部的求解过程，所以划分给了interior point set。

\section*{Remark 2}
在Lemma 8.66中,如果$L_h$如定义般对于$X_\Omega$均有定义,那$L_h$必然不是一个纯粹的离散化椭圆微分算子,会内嵌耦合非Dirichlet的边界条件。这个特性不是很友好，因为(DMP-1)和(DMP-2)理应在这些点上生效，可是在这些非Dirichlet边界点上的限制条件并不是一个正定算子的离散化,而是$au+b\frac{\partial u}{\partial n}=0$类的边界条件的离散化。所以我感觉真在mixed boundray problem中它的应用中可能会有局限,可能(DMP-1,2)不一定能轻易满足。

(目前想到的一种可能的解决方法是干脆把非Dirichlet边界从计算域$\Omega$驱逐出去,直接把线性方程组中的非D边界项消干净再开始。但是消解的过程是否能避免污染系数矩阵的性质还是得再推敲下)

\end{document}
